\(\int x (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x)) \, dx\) [13]

Optimal result

Integrand size = 22, antiderivative size = 120 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=-\frac {5 b d^2 x \sqrt {1+c^2 x^2}}{96 c}-\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2}}{144 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}-\frac {5 b d^2 \text {arcsinh}(c x)}{96 c^2}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2} \]

[Out]

-5/144*b*d^2*x*(c^2*x^2+1)^(3/2)/c-1/36*b*d^2*x*(c^2*x^2+1)^(5/2)/c-5/96*b*d^2*arcsinh(c*x)/c^2+1/6*d^2*(c^2*x
^2+1)^3*(a+b*arcsinh(c*x))/c^2-5/96*b*d^2*x*(c^2*x^2+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5798, 201, 221} \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {5 b d^2 \text {arcsinh}(c x)}{96 c^2}-\frac {b d^2 x \left (c^2 x^2+1\right )^{5/2}}{36 c}-\frac {5 b d^2 x \left (c^2 x^2+1\right )^{3/2}}{144 c}-\frac {5 b d^2 x \sqrt {c^2 x^2+1}}{96 c} \]

[In]

Int[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]

[Out]

(-5*b*d^2*x*Sqrt[1 + c^2*x^2])/(96*c) - (5*b*d^2*x*(1 + c^2*x^2)^(3/2))/(144*c) - (b*d^2*x*(1 + c^2*x^2)^(5/2)
)/(36*c) - (5*b*d^2*ArcSinh[c*x])/(96*c^2) + (d^2*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x]))/(6*c^2)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 5798

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)
^p], Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e
, c^2*d] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {\left (b d^2\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{6 c} \\ & = -\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {\left (5 b d^2\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{36 c} \\ & = -\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2}}{144 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {\left (5 b d^2\right ) \int \sqrt {1+c^2 x^2} \, dx}{48 c} \\ & = -\frac {5 b d^2 x \sqrt {1+c^2 x^2}}{96 c}-\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2}}{144 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2}-\frac {\left (5 b d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{96 c} \\ & = -\frac {5 b d^2 x \sqrt {1+c^2 x^2}}{96 c}-\frac {5 b d^2 x \left (1+c^2 x^2\right )^{3/2}}{144 c}-\frac {b d^2 x \left (1+c^2 x^2\right )^{5/2}}{36 c}-\frac {5 b d^2 \text {arcsinh}(c x)}{96 c^2}+\frac {d^2 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))}{6 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (c x \left (48 a c x \left (3+3 c^2 x^2+c^4 x^4\right )-b \sqrt {1+c^2 x^2} \left (33+26 c^2 x^2+8 c^4 x^4\right )\right )+3 b \left (11+48 c^2 x^2+48 c^4 x^4+16 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{288 c^2} \]

[In]

Integrate[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]

[Out]

(d^2*(c*x*(48*a*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + 3*b*(11 +
48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x]))/(288*c^2)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\frac {d^{2} a \left (c^{2} x^{2}+1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (c x \right )}{96}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 c x \sqrt {c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(116\)
default	\(\frac {\frac {d^{2} a \left (c^{2} x^{2}+1\right )^{3}}{6}+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (c x \right )}{96}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 c x \sqrt {c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(116\)
parts	\(\frac {d^{2} a \left (c^{2} x^{2}+1\right )^{3}}{6 c^{2}}+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{6}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{2}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {11 \,\operatorname {arcsinh}\left (c x \right )}{96}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{36}-\frac {5 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{144}-\frac {5 c x \sqrt {c^{2} x^{2}+1}}{96}\right )}{c^{2}}\)	\(118\)

[In]

int(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^2*(1/6*d^2*a*(c^2*x^2+1)^3+d^2*b*(1/6*arcsinh(c*x)*c^6*x^6+1/2*arcsinh(c*x)*c^4*x^4+1/2*arcsinh(c*x)*c^2*x
^2+11/96*arcsinh(c*x)-1/36*c*x*(c^2*x^2+1)^(5/2)-5/144*c*x*(c^2*x^2+1)^(3/2)-5/96*c*x*(c^2*x^2+1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {48 \, a c^{6} d^{2} x^{6} + 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \, {\left (16 \, b c^{6} d^{2} x^{6} + 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} + 11 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (8 \, b c^{5} d^{2} x^{5} + 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{2}} \]

[In]

integrate(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

1/288*(48*a*c^6*d^2*x^6 + 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b*c^6*d^2*x^6 + 48*b*c^4*d^2*x^4 + 48*
b*c^2*d^2*x^2 + 11*b*d^2)*log(c*x + sqrt(c^2*x^2 + 1)) - (8*b*c^5*d^2*x^5 + 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*s
qrt(c^2*x^2 + 1))/c^2

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.58 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{6}}{6} + \frac {a c^{2} d^{2} x^{4}}{2} + \frac {a d^{2} x^{2}}{2} + \frac {b c^{4} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {b c^{3} d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{36} + \frac {b c^{2} d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {13 b c d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{144} + \frac {b d^{2} x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {11 b d^{2} x \sqrt {c^{2} x^{2} + 1}}{96 c} + \frac {11 b d^{2} \operatorname {asinh}{\left (c x \right )}}{96 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{2}}{2} & \text {otherwise} \end {cases} \]

[In]

integrate(x*(c**2*d*x**2+d)**2*(a+b*asinh(c*x)),x)

[Out]

Piecewise((a*c**4*d**2*x**6/6 + a*c**2*d**2*x**4/2 + a*d**2*x**2/2 + b*c**4*d**2*x**6*asinh(c*x)/6 - b*c**3*d*
*2*x**5*sqrt(c**2*x**2 + 1)/36 + b*c**2*d**2*x**4*asinh(c*x)/2 - 13*b*c*d**2*x**3*sqrt(c**2*x**2 + 1)/144 + b*
d**2*x**2*asinh(c*x)/2 - 11*b*d**2*x*sqrt(c**2*x**2 + 1)/(96*c) + 11*b*d**2*asinh(c*x)/(96*c**2), Ne(c, 0)), (
a*d**2*x**2/2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (104) = 208\).

Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.95 \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{6} \, a c^{4} d^{2} x^{6} + \frac {1}{2} \, a c^{2} d^{2} x^{4} + \frac {1}{288} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{2} \, a d^{2} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{2} \]

[In]

integrate(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

1/6*a*c^4*d^2*x^6 + 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arcsinh(c*x) - (8*sqrt(c^2*x^2 + 1)*x^5/c^2 - 10*sqrt(c^
2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/c^6 - 15*arcsinh(c*x)/c^7)*c)*b*c^4*d^2 + 1/16*(8*x^4*arcsinh(c*x)
- (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*b*c^2*d^2 + 1/2*a*d^2*x^2
+ 1/4*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*b*d^2

Giac [F(-2)]

Exception generated. \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]

[In]

int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^2,x)

[Out]

int(x*(a + b*asinh(c*x))*(d + c^2*d*x^2)^2, x)